The square \(G^2\) of a graph \(G\) is the graph defined on \(V(G)\) such that two vertices \(u\) and \(v\) are adjacent in \(G^2\) if the distance between \(u\) and \(v\) in \(G\) is at most 2. Let \(\chi(H)\) and \(\chi_{\ell}(H)\) be the chromatic number and the list chromatic number of \(H\), respectively. A graph \(H\) is called \({chromatic-choosable}\) if \(\chi_{\ell} (H) = \chi(H)\). It is an interesting problem to find graphs that are chromatic-choosable.

Motivated by the List Total Coloring Conjecture, Kostochka and Woodall (2001) proposed the List Square Coloring Conjecture which states that \(G^2\) is chromatic-choosable for every graph \(G\). Recently, Kim and Park showed that the List Square Coloring Conjecture does not hold in general by finding a family of graphs whose squares are complete multipartite graphs with partite sets of unbounded size. It is a well-known fact that the List Total Coloring Conjecture is true if the List Square Coloring Conjecture holds for special class of bipartite graphs. On the other hand, the counterexamples to the List Square Coloring Conjecture are not bipartite graphs. Hence a natural question is whether \(G^2\) is chromatic-choosable or not for every bipartite graph \(G\).

In this paper, we give a bipartite graph \(G\) such that \(\chi_{\ell} (G^2) \neq \chi(G^2)\). Moreover, we show that the value \(\chi_{\ell}(G^2) – \chi(G^2)\) can be arbitrarily large. This is joint work with Boram Park.

The well-known Ohba Conjecture says that every graph G with |V(G)|≤ 2χ(G)+1 is chromatic choosable.
This paper studies an on-line version of Ohba Conjecture. We prove that unlike the off-line case, for k≥3, the complete multipartite graph K2*(k-1), 3 is not on-line chromatic-choosable. Based on this result, the on-line version of Ohba Conjecture is modified as follows: Every graph G with |V(G)|≥ 2χ(G) is on-line chromatic choosable. We present an explicit strategy to show that for any positive integer k, the graph K2*k is on-line chromatic-choosable. We then present a minimal function g for which the graph K2*(k-1), 3 is on-line g-choosable. This is joint work with Young Soo Kwon, Daphne Der-Fen Liu, and Xuding Zhu.

Say that a graph with maximum degree at most d is d-bounded. For d>k, we prove a sharp sparseness condition for decomposability into k forests and a d-bounded graph. Consequences ar e that every graph with fractional arboricity at most k+ d/(k+d+1) has such a decomposition, and (for k=1) every graph with maximum average degree less than 2+2d/(d+2) decomposes into a forest and a d-bounded graph. When d=k+1, and when k=1 and d≤6, the d-bounded graph in the decomposition can also be required to be a forest. When k=1 and d≤2, the d-bounded forest can also be required to have at most d edges in each component.
This is joint work with A.V. Kostochka, D.B. West, H. Wu, and X. Zhu.

An injective coloring of a graph G is an assignment of colors to the vertices of G so that any two vertices with a common neighbors receives distinct colors. The injective chromatic number, \(\chi_i(G)\), is the minimum number of colors needed for an injective coloring. Let \(mad(G)\) be the maximum average degree of G. In this paper, we show that \(\chi_i(G)\leq\Delta + 2\) if \(\Delta(G) \geq 4\) and \(mad(G) \leq \frac{14}{5}\). When \(\Delta(G) = 3\), we show that \(\)mad(G) < \frac{36}{13}[/latex] implies [latex]\chi_i(G) \leq 5[/latex]. This is sharp; there is a subcubic graph [latex]H[/latex] such that [latex]mad(H) = \frac{36}{13}[/latex], but [latex]\chi_i(H) = 6[/latex]. This is joint work with Daniel Cranston and Gexin Yu.